Construction of low-density parity-check codes from Kirkman triple systems

Gallager introduced LDPC codes in 1962, presenting a construction method to randomly allocate bits in a sparse parity-check matrix subject to constraints on the row and column weights. Since then improvements have been made to Gallager’s construction method and some analytic constructions for LDPC codes have recently been presented. However, analytically constructed LDPC codes comprise only a very small subset of possible codes and as a result LDPC codes are still, for the most part, constructed randomly. This paper extends the class of LDPC codes that can be systematically generated by presenting a construction method for regular LDPC codes based on combinatorial designs known as Kirkman triple systems. We construct -regular codes whose Tanner graph is free of -cycles for any value of divisible by .